Step 1:
Following the arguments laid out in Section , (also see [1]}), in Step  of Algorithm , we choose the special basis functions \(\lbrace \Psi _n\rbrace _{n \ge 1}\)  of the space \(L^2(0,T)\) , which was originally introduced by Klibanov in [2]}.
We recall that Klibanov's basis is obtained by applying the Gram-Schmidt orthogonalization process for the sequence of functions \(\lbrace \Phi _n(t)\rbrace _{n\ge 1}\)  where \(\Phi _n(t)=t^{n-1} e^{t-T/2}\) . A comparative advantage of this special basis over traditional trigonometric Fourier basis is that the first term of the trigonometric basis being a constant, vanishes on taking the derivatives.
As such, there will be no contribution from the coefficient of the first basis function on the left hand side of equation (REF ), i.e. the information from \(u_1(x)\Psi _1^{\prime }(t)\)  is lost, while this information is retained if we use Klibanov's basis. To choose the cut off number \(N\) ,
we take the data \(u({\bf x}, t)\)  for \({\bf x}\in \Gamma \)  and \(t \in (0, T)\)  where
\(\Gamma = \lbrace {\bf x}= (x, y = R): |x| \le R\rbrace  \subset \partial \Omega .\) 
